# Thread: BMO 2006/7 circles question

1. ## BMO 2006/7 circles question

Two touching circles S and T share a common tangent which meets
S at A and T at B. Let AP be a diameter of S and let the tangent
from P to T touch it at Q. Show that AP = PQ.

If you have solved it then please give me a hint(don't post the full solution).
Thanks in advance.

2. ## Re: BMO 2006/7 circles question

Originally Posted by abhishekkgp
Two touching circles S and T share a common tangent which meets
S at A and T at B. Let AP be a diameter of S and let the tangent
from P to T touch it at Q. Show that AP = PQ.

If you have solved it then please give me a hint(don't post the full solution).
Thanks in advance.
1. Draw a sketch!

2. Prove(!) that the described situation is only possible if the 2 circles are congruent.

3. Prove that the 2 diameters and the tangent segments consequently form a square.

3. ## Re: BMO 2006/7 circles question

Originally Posted by earboth
1. Draw a sketch!

2. Prove(!) that the described situation is only possible if the 2 circles are congruent.

3. Prove that the 2 diameters and the tangent segments consequently form a square.
I have attached a sketch. The two circle S and T drawn are not congruent but i still get AP=PQ.